In 1997, the state of Ohio changed its requirements for teacher certification. These changes required the creation of two new programs: a pre-K through 3 program and a 4-9 program. While the mathematics and mathematics education departments discussed these programs, it was decided to also address the secondary education program. Both departments wanted to see changes in the secondary program. The mathematics education department wanted to remove a troublesome requirement, and the mathematics department wanted to add three hours of content material to the old requirements. A compromise was struck where students were allowed to take a mathematics capstone course in place of the previous requirement, but also adding three hours of mathematics content requirements. I was chosen to design the capstone course, with the understanding that the course should address the connection between collegiate mathematics and high school mathematics in some way. After many discussions with others about what such a course might look like, I chose the topic of the creation and understanding of the real numbers to be the theme of the course.
I first ran the course in the fall term of 1998 at Bowling Green State University. I had 10 students in the class. In addition to the usual homeworks, midterms, and exams, I required each student to choose one week of the class for which they would be responsible for writing up course notes for the class. That term, the basic content covered:
The notes the students generated were then put together with some notes that I wrote myself, and these became the textbook for the following terms.
I felt that the course went well for a first time, but I was aware that I had a lot of work to do to make the course better. Two particular thoughts came to my mind: while the students in the course were excellent, I did not feel that they had any experience of actually doing mathematics as a mathematician might, and that I did not do as good a job as I should in tying topics to the high school curriculum.
The next two times I ran the course, I used the notes generated in the first term. However, for the second time, I decided to address the issue of having the students do mathematics by assigning semester-long research projects to teams of students. The idea of assigning such projects stemmed from discussions with my colleague David Meel, an expert in mathematics education. At this point, I needed to come up with some projects, and I did so by scouring the mathematics literature and colleagues for ideas of interesting mathematics problems that had classical issues arising in them. By this, I mean that either they led to one of the classical numbers or topics in mathematics. In particular, I looked for problems in which e and pi showed up somewhat unexpectedly. (For a partial list of the early projects, click here.)
The second running of the course went fairly well, except that I received some of the worst student evaluations I have ever received (although I was still rated better than average). Some students felt that the course did not do a good job addressing the high school curriculum, and they were angered by the amount of work the class required. There were exceptions to this. The most interesting of which, was that I had one graduate student in the class that had taught high school for several years. She mentioned to her advisor (in the education department) that she could see how what I was doing was tying together things from the high school curriculum, but that for some reason, the other students couldn't see it. Many of the best students (the evaluation forms at BGSU have students put down their expected grade) did see some reason behind the class, but the weaker students did not.
In the third term that the course ran, several things happened. First, because I had taken on new administrative responsibilities in the department, my teaching load was reduced. For a variety of reasons, the capstone course was the course that I was going to stop teaching. However, the course was not yet ready for me to hand over, so in the fall term of 1999, David Meel and I taught the course together. We kept the projects with some modifications, however, we worked together to make the course tie in to the high school curriculum in a fashion that all the students could see more obviously.